Distributive extensions of modules

Let X be a submodule of a module M. The extension is said to be distributive if X ∩ (Y + Z) = X ∩ Y + X ∩ Z for any two submodules Y and Z of M. We study distributive extensions of modules over not necessarily commutative rings. In particular, it is proved that the following three conditions are equivalent: (1) is a distributive extension; (2) for any submodule Y of the module M, no simple subfactor of the module X/(X∩Y ) is isomorphic to any simple subfactor of Y/(X∩Y) (3) for any two elements x ∈ X and m ∈ M, there does not exist a simple factor module of the cyclic module xA/(X ∩ mA) that is isomorphic to a simple factor module of the cyclic module mA/(X ∩ mA). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 141–150, 2006.     

On extensions of rational modules

Given a topological algebra A, we investigate when the categories of all rational A-modules and of finite-dimensional rational modules are closed under extensions inside the category of A-modules. We give a complete characterization of these two properties, in terms of a topological and a homological condition, for complete algebras. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes of coalgebras for which rational modules are closed under extensions as well as various examples where this is not true.     

Minimal silting modules and ring extensions

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behavior of minimality under ring extensions. We show that minimal cosilting modules over a commutat...     

Gorenstein projective modules and Frobenius extensions

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x~2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.     

Fuzzy preorders: conditional extensions,extensions and their representations

The crisp literature provides characterizations of the preorders that admit a total preorder extension when some pairwise order comparisons are imposed on the extended relation. It is also known that every preorder is the intersection of a collection of total preorders. In this contribution we generalize both approaches to the fuzzy case. We appeal to a construction for deriving the strict preference and the indifference relations from a weak preference relation, that allows to obtain full characterizations in the conditional extension problem. This improves the performance of the construction via generators studied earlier.     

Tilting modules over split-by-nilpotent extensions

Let a finite dimensional algebra R be a split extension of an algebra A by a nilpotent bimodule Q. We give necessary and sufficient conditions for a (partial) tilting module T_A to be such that T?_A R_R is a (partial) tilting module. If this is not the case, but Q_A is generated by the tilting module T_A , then there exists a quotient [Rbar] of R such that T?_A [Rbar]_[Rbar] is a tilting module.     

Gluing representations via idempotent modules and constructing endotrivial modules

Let G be a finite group and k be a field of characteristic p. We show how to glue Rickard idempotent modules for a pair of open subsets of the cohomology variety along an automorphism for their intersection. The result is an endotrivial module. An interesting aspect of the construction is that we end up constructing finite dimensional endotrivial modules using infinite dimensional Rickard idempotent modules. We prove that this construction produces a subgroup of finite index in the group of endotrivial modules. More generally, we also show how to glue any pair of kG-modules.     

Jacquet modules of ladder representations

We compute the Jacquet modules for a certain class of irreducible representations of the general linear group over a non-Archimedean local field. This class contains the Speh representations.     

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.     

Examples of rings and modules as trivial extensions

Abstract

The two main classes of finite dimensional composition algebras, Hurwitz and symmetric, are characterized in terms of the flexible identity, and even of the third power associative identity, over fields of arbitrary characteristic.